Abstract
Let P be a probability distribution on ∝d and let \(C\) be the family of the uniform probabilities defined on compact convex sets of ∝d with interior non-empty. We prove that there exists a best approximation to P in \(C\), based on the L 2-Wasserstein distance. The approximation can be considered as the best representation of P by a convex set in the minimum squares setting, improving on other existent representations for the shape of a distribution. As a by-product we obtain properties related to the limit behavior and marginals of uniform distributions on convex sets which can be of independent interest.
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Cuesta-Albertos, J.A., Matrán, C. & Rodríguez-Rodríguez, J. Approximation to Probabilities Through Uniform Laws on Convex Sets. Journal of Theoretical Probability 16, 363–376 (2003). https://doi.org/10.1023/A:1023518526754
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DOI: https://doi.org/10.1023/A:1023518526754